Generalised Monge–Ampère equations, introduced by Pingali, include several well-known PDE on compact Kähler manifolds, such as the J-equation, inverse Hessian equations, and certain deformed Hermitian–Yang–Mills equations. By results of Datar–Pingali and Fang–Ma, extending work of Gao Chen, Song, and others, solvability is equivalent to a Demailly–Păun type positivity condition tested on all irreducible analytic subvarieties; I will discuss when this infinite collection of tests can be reduced to a finite set of distinguished subvarieties, yielding effective criteria for solvability in several settings. The main results discussed are based on joint work with Sohaib Khalid, with recent ongoing developments joint with Sivaram Petchimuthu on optimal destabilising subvarieties and singularity formation of the J-flow on threefolds.